Elixirs refers to one of four different types of potion in Diablo, the elixirs of Strength, Dexterity, Magic and Vitality.
Once the player gets far enough in the game, they can buy elixirs for 5000 gold each. Drinking one of these will permanently raise it's corresponding stat by one point. A stat cannot go over it's maximum in this way.
Elixirs of vitality are the rarest elixir, and cannot be bought in stores.